A toughie trigonometric integral

Prove that

    \[\int_{0}^{\infty} \frac{\sin^2 \tan x}{x^2} \, \mathrm{d}x = \frac{\pi}{2}\]

Solution

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An Euler squared sum

Let \mathcal{H}_n denote the n -th harmonic number. Prove that

    \[\sum_{n=1}^{\infty} \left ( \frac{\mathcal{H}_n}{n} \right )^2 = \frac{17 \pi^4}{360}\]

Solution

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Not Lebesgue integrable function

Let x \in \mathbb{R}. Given the series

(1)   \begin{equation*} \sum_{n=2}^{\infty} \frac{\sin nx}{\ln n} \end{equation*}

  1. Prove that (1) converges forall x \in \mathbb{R}.
  2. Prove that (1) is not a Fourier series of a Lebesgue integrable function.

Solution

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